Finding large H-colorable subgraphs in hereditary graph classes

We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: in {P5, F}-free graphs in polynomial time, whenever F is a threshold graph; in {P5, bull}-free graphs in polynomial time; in P5-free graphs in time n^O(ω(G⁾⁾; in {P6, 1-subdivided claw}-free graphs in time n^O(ω(G⁾³⁾ . Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6, 1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs, if we allow loops on H.